I'm doing a critique of an Economics paper, and I want to have my critique about omitted variable bias. I have found a variable that was not included that is both correlated with my regressor of interest (The covariance does not equal 0), and is a determinant of the dependent variable (its coefficient is significantly different from 0). However, by including this omitted variable, it causes the coefficient of my regressor of interest to become statistically insignificant at the 5% level. Whereas, the coefficient on my regressor of interest was significant at the 5% level before the omitted variable was included.

What does this mean? And is this still omitted variable bias given the change in significance?

1 Answer
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Typically, you have to combine statistical inference with economic theory or intuition if you want to make a strong claim about omitted variable bias. In small samples, regression coefficients are always going to be sensitive to the addition or subtraction of variables and this would be an easy critique to make in that you have already shown that the result presented is sensitive to the model specification.

However, there are many ways in which adding a variable can change the significance and thus it is not easy to say this is certainly omitted variable bias. For example, the variable you added could be a stronger proxy for the same underlying phenomenon. Or you could be introducing collider bias or bias amplification for example. This is where you need to put on your economist hat and consider the structure of what you are trying to model.